# How many solutions does this system have? {9x + 3y = 2 {y = -3x + 5/3 a) 1 solution b) 2 solutions c) no solutions d) infinite solutions

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Solve the system: `9x + 3y = 2`

`y = -3x + 5/3`

First, multiply 2nd equation by 3 nd rewrite in `ax + by =c` form.

This gives us: `3y = -9x + 5`

`-9x - 3y = -5`

Using the elimination method, add the 2 equations together.

`9x + 3y = 2`

`-9x-3y = -5`

`0 = -3`

Since `0!=-3,`

there is no solution.

**Therefore, choice "c - no solutions" is the correct solution.**

The slopes and y-intercepts of the two given lines can also determine the number of solutions, given in y=mx+b form. (m, the slope of the line and b the y-intercept). Graphically speaking, the point of intersection is the solution to a system. Given one solution to the system means that the line would have different slopes and perhaps a different y-intercept so that the lines could intersect. On the other hand, 2 solutions are most often found given in much more complex systems (quadratic systems). No solution results from lines that are parallel to each other, hence the slope would be the same, but must have **different** y- intercepts. Lastly, infinite number of solutions is the result of two equivalent equations, as it would graphically appear right on top of each other. Therefore, the lines consist of the same slope and y-intercepts.

Here are the following steps to guide you through this particular question:

1) Change any equations that are not in the form of y=mx+b, in this case, 9x + 3y = 2.

2)9x + 3y = 2

y = -3x + 2/3

3) Compare the slopes and y-intercepts of both lines.

As the m-value or the slope (-3) is the same, this means the there would be** no solutions** to this system. Therefore, **c** is the correct answer.